Solve ∫ (from 1 to 2) of x^3d

$$\int _ { 1 } ^ { 2 } x ^ { 3 } d$$

Evaluate

$$\frac{15d}{4}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{3}d\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$d\int x^{3}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}.

$$d\times \frac{x^{4}}{4}$$

Simplify.

$$\frac{dx^{4}}{4}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{4}d\times 2^{4}-\frac{1}{4}d\times 1^{4}$$

Simplify.

$$\frac{15d}{4}$$

Differentiate w.r.t. d

$$\frac{15}{4} = 3\frac{3}{4} = 3.75$$